Aitken-Steffensen-type methods for nonsmooth functions (I)

We study the convergence of the Aitken-Steffensen method for solving a scalar equation \(f(x)=0\). Under reasonable conditions (without assuming the differentiability of \(f\)) we construct some auxilliary functions used in these iterations, which generate bilateral sequences approximating the solution of the considered equation.

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

nonlinear equations in R; Aitken-Steffensen method.

PDF-LaTeX file on the journal website.

I. Păvăloiu, Aitken-Steffensen-type methods for nonsmooth functions (I), Rev. Anal. Numér. Théor. Approx., 31 (2002) no. 1, pp. 109-114.

Balázs, M., A bilateral approximating method for finding the real roots of real equations, Rev. Anal. Numér. Théor. Approx., 21, no. 2, pp. 111-117, 1992, http://ictp.acad.ro/jnaat/journal/article/view/1992-vol21-no2-art3

Casulli, V. and Trigiante, D., The convergence order for iterative multipoint procedures, Calcolo, 13, no. 1, pp. 25-44, 1977, https://doi.org/10.1007/BF02576646

Cobzaş, S., Mathematical Analysis, Presa Universitară Clujeană, Cluj-Napoca, 1997 (in Romanian).

Ostrowski, A. M., Solution of Equations and Systems of Equations, Academic Press, New York, 1960.

Păvăloiu, I., On the monotonicity of the sequences of approximations obtained by Steffensens’s method, Mathematica (Cluj), 35 (58), no. 1, pp. 71-76, 1993.

Păvăloiu, I., Bilateral approximations for the solutions of scalar equations, Rev. Anal. Numér. Théor. Approx., 23, no. 1, pp. 95-100, 1994, http://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art10

Păvăloiu, I., Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequences, Calcolo, 32, no. 1-2, pp. 69-82, 1995, https://doi.org/10.1007/BF02576543

Traub, F. J., Iterative Methods for the Solution of Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.